急 ! F4 Quadratic Equations

15.
y = ax^2 - 6x + 5

if the equation above does not intersect the x-axis
There will be no solution when y = 0
Such that Δ < 0

Δ = b^2 - 4ac
0 > (-6)^2 - 4a(5)
0 > 36 - 20a
20a > 36
a > 1.8

If a is an integer, the minimum value would be 2.


17.
(x + 1)^2 = 4x
x^2 + 2x + 1 - 4x = 0
x^2 - 2x + 1 = 0

Δ = (-2)^2 - 4
Δ = 4 - 4 = 0

The equation has two equal real roots


18.
2(x + 1)^2 - 8(x + 1) - 1 = 0
2x^2 + 4x + 2 - 8x - 8 - 1 = 0
2x^2 - 4x - 7 = 0

y = 2(x + 1)^2 - 8(x + 1) - 1
y = 2x^2 - 4x - 7

The equation intersects the x-axis when y = 0
such that 2x^2 - 4x - 7 = 0

Δ = (-4)^2 - 4(2)(-7)
Δ = 16 + 56
Δ = 72

Since Δ > 0
There are two points of intersection between y = 2(x + 1)^2 - 8(x + 1) - 1 and the x-axis.


19.
(1 + 2m)x^2 - 4x - 2 = 0

If the equation above has two distinct real roots, Δ > 0

Δ = (-4)^2 - 4(1 + 2m)(-2)
0 < 16 + 8 + 4m
4m > -24
m > -6


20.
4(x^2 - 3x) + k = 3
4x^2 - 12x + k - 3 = 0

If the equation above has no real root, Δ < 0

Δ = (-12)^2 - 4(4)(k - 3)
0 > 144 - 16k + 48
16k > 192
k > 12


21.
2x^2 - 4x + (c - 1) = 0

If the equation above has real root, Δ >= 0

Δ = (-4)^2 - 4(2)(c - 1)
0 <= 16 - 8c + 8
8c <= 24
c <= 3